On the Intensity of Aeolian Sound under Water

LETTERS TO THE EDITOR

Received 21 April 1972 13.7; 13.8; 13.9

On the Intensity of Aeolian Sound under Water

H. CASTELLIZ

Hermes Electronics Ltd. (formerly EMI Cossor), Dartmouth, Nova Scotia, Canada

The study aims at prediction of the intensity of Aeolian sound created under water by an ocean current passing a wire or cable. Particular application is a mooring line of a deep-sea taut-moored buoy. The work is based on a paper by Phillips [J. Fluid Mech. 1, 607-624 (1956)] but the difference in scale between sound waves in air and in water necessitates the extension of Phillips' equations into the acoustical nearfield. A brief outline of the theory and numerical values are given.

INTRODUCTION

This note deals with the intensity of Aeolian sound generated under water. It assumes a wire or cable suspended in the ocean and exposed to ocean currents. The aim is to obtain an estimate of the amplitude of the radiated sound at the Strouhal frequency, and in the vicinity of the wire.

The theory of Aeolian sound is well established (Lighthill, 1 Curle2); but prediction of the intensity requires knowledge of certain numerical constants which can only be determined experimentally. Such measurements, made by several authors, are reviewed by Phillips 3 in a most valuable study on Aeolian tones. Based on this work, numerical prediction is possible.

However, there is a certain difference in scale between the sound fields of Aeolian sound under water and in

air which necessitates a separate treatment, as most papers dealing with the subject do so with applications in air in view.

In underwater applications it is of interest to know the Aeolian sound level in the relative vicinity of the generating wire, that is, at distances less than the acoustical wavelength. This note, therefore, amends Lighthill's theory by expanding it into the acoustic nearfield region.

It is necessary to discriminate between acoustic and hydrodynamic "vicinity." Lighthill's theory is based on the replacement of the actual sound source (the eddies) by an acoustic dipole and therefore leads to correct results only at distances from the source which are "great" hydrodynamically, i.e., relative to the down- stream extension of the eddy field. In air, this usually also implies distances "great" acoustically, relative to the wavelength; the spatial periodicities of the eddies and of the sound are of the same order of magnitude.

In water, this is different. Here, the kinematic vis- cosity being ten times less and the velocity of sound five times greater than in air, the wavelength of an Aeolian tone is •50 times greater, for the same Reynolds number and same wire diameter. Hence, a point hydrodynamically "far" may well be "near" acoustically, and the nearfield term in the sound equation becomes important.

I. OUTLINE OF THEORY

According to the theory of Lighthill and Curle, the sound created by the flow around a solid incompressible object is such as would be created by a fluctuating force acting on the body and via the body on a station- ary fluid, where this force is equal (and opposite) to that which, in the real case, the fluid exerts onto the object. The body, in the fi.ctitious case, is assumed of the same mass density as the fluid.

The sound source equivalent to a line element of the wire thus may be considered a "dipole" whose strength is equal to the hydrodynamic lift. The lift is perpendicular to the flow and to the wire and so, therefore, is the dipole axis.

The theory has been tested experimentally, in particular by Powell 4 for the related case of an edge tone. He measured the "lift" and the sound output simul- taneously and confirmed convincingly the basic theo- retical statement.

The fundamental difficulty, however, lies in determination of the lift from the hydrodynamic conditions. The Kutta-Joukowsky equation for the lift of a body in a fluid flow is not applicable under alternating conditions and would lead to much too high lift values. This is one of the points where experimental results must be used.

The other point is the determination of the correlation length. Assume any finite length of wire. The eddy formation will not occur coherently over any appreciable length of the cylinder. Due to irregularities in the flow and/or the wire, a phase shift will occur in the alternating lift between any two points on the wire which are separated by more than a certain small distance, the "correlation length" Only within the correlation length, roughly speaking, the process is con- sistent in phase. Thus, the sound pressure originating from a wire will be proportional to wire length as long as the length is less than the correlation length. For longer wires, however, the individual elements (of one correlation length each) will contribute randomly to the field, and the sound intensity will be additive. There- fore, for any appreciable wire length, the mean square of the sound pressure will be proportional to the length.

1062 Volume 52 Number 3 (Part 2) 1972

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OBSERVER

z=O

FIG. 1. Geometry of the problem.

DIRECTION OF FLOW

CYLINDER ELEMENT

DIRECTION OF LIFT (DIPOLE AXIS)

Both the influence of the correlation length and the value of the lift result in a common numerical factor in

the final sound equation. This factor has been determined by Phillips a for a range of Reynolds numbers (see below).

Briefly, the derivation may be sketched as follows. The pressure field of an acoustical dipole is

p(r,O)= (F/4•rr2)(lq-ikr)cosOXe i(•t-kr), (1)

where F is the dipole strength, or, in our application, the amplitude of the "lift"; co is its angular frequency, in our case given by the Strouhal frequency; r the distance from dipole center to the point of observation; 0 the angle subtended between dipole axis and r (the geometry is illustrated in Fig. 1); t is time; and k = 2•r/X =co/c, c being the speed of sound in the medium. If F is taken per unit length of cable, then p is the pressure increment contributed "per unit length of cable."

According to the above consideration, the contribu- tion of one correlation length, say sd, (d-diameter of the wire) will be Eq. 1 multiplied by sd, i.e., p(r,O)sd. In order now to find the sound originating from a length

L greater than sd, we calculate the mean square of psd, and multiply by the number of correlation lengths contained in L, L/sd. Subsequent division by L yields a value for the mean-square pressure (p'2)av, "per unit length," as a basis for integration if the field radiated from a length >>sd is to be calculated, where r and 0 may vary over the integration length. It results in

(p'"•.,r = (p•sd),,= [-F•d/2(4,rr•2](1 +k••cos•. (2)

Phillips shows that the correlation length is about ! 7 diameters for Reynolds numbers between 100 and 150 and still less for higher Reynolds numbers. Therefore, the above condition (r>>sd) is fulfilled in all cases of practical interest. We assume here that the cable does not vibrate (see below); if it would, the correlation distance might become much greater due to a feedback mechanism forcing the eddies into a phase-coherent pattern over considerable stretches (cf. galloping telegraph wires).

The lift amplitude F may be expressed in the general form

F = '•pd U •, (3)

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where o is the density of the fluid and U the velocity of the flow. The factor • must be determined by experi- ment and values are given by Phillips.

The frequency of the eddy formation and thus the acoustical frequency is given by

f =St//•, (4)

where $ is the Strouhal number.

Introducing Eqs. 3 and 4 into Eq. 2, and considering k=2•rf/c, lead to the final expression for the mean- square pressure per unit length of cable.

(p, 2av(r,O) •2UøS2 l_}_l• =, -- cos"O, (5) c 2 \r • k24/

where

a=?=/8 (6)

is a constant into which the numerical factors for lift and scale are subsumed.

We are interested in the total sound pressure P in a point A located far enough from the wire to assure R to be large relative to the flow pattern and to the correlation length, but not necessarily far acoustically, so that for a stretch of the wire the acoustic nearfield

may become relevant. Integration of Eq. 5 over a given wire length between

two points z• and z=is achieved by elementary methods after expressing the variables r and 0 in terms of the constants R and • and the integration variable z. (R is the perpendicular distance observer to wire, • the azimuth of the receiver relative to the direction of the

dipole axis, and z is the coordinate along the wire axis, with z=0 at the foot of the perpendicular from observer to the wire. See Fig. 1.)

It results for the mean-square pressure that

( P••., =a (o• Uø3• 2c • cos% X [-//(z, --//(z x) + (1/2k " (G (z, -O(z•))-I,

where//and G are defined by the expressions

and

(7)

tt(z)=•z/(R2z2)]+R -• arc tan(z/R) (8)

G(z) = [-z/ (R2}-z221+ (3/2R2tt (z). (9)

They determine the farfield and nearfield contributions, respectively.

II. NUMERICAL VALUES

Phillips a shows that, for Reynolds numbers between about 50 and 150, the correlation length is approxi-

mately !7 diameters; for the same range, the lift factor is given as •=0.38. Consequently, in this range (non- turbulent wake) the factor a=0.31 results. 5

For higher Reynolds numbers, both s and • will be less, and Phillips deduces from experimental results obtained by two other authors that a value a=0.037 fairly well covers the range of Reynolds numbers from 360 to 30 000.

As mentioned above, these values are obtained under conditions where the wire does not vibrate. With ten-

sioned wires of limited length, which possess transverse- wave resonances, flow-induced vibrations (strumming) may occur. However, if the wire is very long, e.g., a mooring line of a taut-moored, deep-sea buoy, vibration will not occur because transverse standing waves cannot develop; the wavelength would be in the order of meters and any traveling wave excited somewhere would be damped out before it would be reflected from the far end.

Based on this argument, the following example may be calculated to illustrate the sound intensity to be expected.

Assume that the daimeter of wire, d=0.3 cm; the length of wire:semi-infinite, zx =0, z•.= •; the distance of hydrophone from wire, R= 180 cm; the azimuth of hydrophone, referred to dipole axis, •=0; the velocity of current, U=3 knots= 150 cm/sec.

It follows that the Reynolds number, R=3X10a; the Strouhal number, S=0.2; the vortex shedding frequency, f=100 Hz; the wavelength in water, X = 1500 cm; and the factor a=0.037.

From these figures it follows that the rms value of the sound pressure (Eq. 7) at the location of the hydrophone is

Prra,=0.046 #bar or -27 dB re 1 #bar.

For comparison, the minimum sea noise, at the same frequency, in low traffic conditions in the deep sea, 6 in 1-Hz bandwidth, is approximately-32 dB re ! #bar.

• M. J. Lighthill, Proc. Roy. Soc. (London) 3.211, 564-587 (1952).

"N. Curle, Proc. Roy. Soc. (London) A231, 505-514 (1955). a O. M. Phillips, "The Intensity of Aeolian Tones," J. Fluid

Mech. 1, 607-624 (1956). • A. Powell, J. Acoust. Soc. Amer. 33, 395409 (1961). * Phillips has a=0.27 in his Eqs. (6.1) and (6.2), but following

his calculations one can easily show this to be a misprint. 6 G. M. Wenz, J. Acoust. Soc. Amer, 34, 1936-1956 (1962).

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On the Intensity of Aeolian Sound under Water